Dynamics of Bloch Electrons in Time Dependent External Electric Fields: Bounds for Interband Transitions
aa r X i v : . [ m a t h - ph ] S e p Dynamics of Bloch Electrons in TimeDependent External Electric Fields: Boundsfor Interband Transitions
A. NenciuFaculty of Applied Sciences University “Politehnica” of Bucharest,Splaiul Independentei 313, RO-060042 Bucharest, Romania
Abstract
Using adiabatic expansions formalism, upper bounds for interbandtransitions for Bloch electrons in slowly varying in time electric fieldsare obtained. These bounds imply the validity of one-band approxi-mation on long time scales.
This paper is devoted to the generalization of the main result in [1] concerningthe smallness of the interband transitions for homogeneous time independentexternal electric fields to slowly time dependent electric fields. The study ofBloch electrons in a time independent electric field has a long and distin-guished history. The subject is as old as the quantum theory of solids (seee.g. [2] for an extensive discussion) but, as the problem of the interbandtransitions is concerned, the real story started with the papers of Wannier[3], [4] who argued that in the presence of a weak homogeneous time inde-pendent electric field the energy bands of the crystal are ”deformed” andthere are no interband transitions between the deformed bands. Moreover,the Hamiltonian restricted to a simple deformed band consists of a ladderof discrete eigenvalues (Stark-Wannier ladder). Wannier claims were chal-lenged by Zak [5] on the ground that in the presence of arbitrarily weakfield the spectrum becomes continuous so Stark-Wannier ladders of boundstates cannot exist and indeed, it has been rigorously proved (see e.g. [6],[7]) that for sufficiently regular periodic potentials (for singular, e.g. δ -likepotentials, the situation might be different; see [8], [9] and the references1herein) the spectrum is absolutely continuous in the presence of a weak ho-mogeneous time independent electric field so, if Stark-Wannier ladders exist,they consist of resonances. The issue remained controversial for decades andeventually settled down in the affirmative at the rigorous level by using pow-erful mathematical tools (for references and a detailed discussion see sectionsIA, IV and VIA in [2]). One of the key steps was the proof in [1] that onecan define recurrently deformed bands for which the interband transitionsare smaller than any power of the electric field strength. In its time inde-pendent form the expansion method in [1] has been considerably extended in[10], [11]. Considered initially as an interesting but academic problem, theexistence of Stark-Wannier ladders of resonances was experimentally provedafter the invention of superlattices (see [12] and the references therein) andeven more, found technological applications (see e.g. [13]).Since the time independent electric fields are (ideal) limits of slowly vary-ing in time electric fields it is naturally to try to extend the whole analysis toslowly varying fields. At the heuristic level one expects by an adiabatic argu-ment that the interband transitions are still small and one can hope to provethe same type of result about the existence of almost invariant deformedbands. Such a generalization was conjectured already in [1] and indeed, in[14], [15] we developed a similar theory as in the time independent case up tothe second order. Unfortunately, for higher orders the computations becomeunmanageably complicated.In this paper we shall develop a different procedure based on the adiabaticexpansion in [16] which allows us to push the construction of the deformedbands for slowly varying in time electric fields to arbitrary order.The content of the paper is as follows: Section 2 contains a brief review ofthe result in [1] about time independent case, the description of the problemand the main result. Section 3 contains the construction of the orthogonalprojection on the subspaces describing the deformed bands. Finally, Section4 contains the proofs. We begin with a short review of the main result in [1]. For simplicity we shalltreat one-dimensional case, but the results are valid for arbitrary dimensions.The Hamiltonian describing one electron subjected to a periodic potentialand to a perturbation given by a homogeneous time independent eletric field E is: H ε = H + εX (2.1)2here ε = − eE ; H = − m d dx + V ( x ); V ( x + na ) = V ( x ) (2.2)and a is the lattice constant.The spectrum of H , σ ( H ) = σ , is supposed to have at least one isolatedband σ separated by the rest of the spectrum: σ = σ ∪ σ dist ( σ , σ ) = d > E , the potential energy goes to infinity atlarge distances and the ordinary perturbation theory cannot be applied. TheHamiltonian of the perturbed system can be written in the following form: H ε = P H ε P + (1 − P ) H ε (1 − P ) + ( P H ε (1 − P ) + h.c. )where P is the orthogonal projection on the subspaces of states correspond-ing to the isolated band σ of H . As already remarked by Callaway [17],[18], the one-band Hamiltonian P H ε P has a discrete spectrum called Stark-Wannier ladder of the form α + εak , where α is a constant, a the crystalconstant and k an integer. As for in band dynamics, the electron is notcontinuous accelerated, but will undergo a periodic motion in k -space causedby the Bragg reflections at the boundary of the Brillouin zone, having theperiod T = πεa . This oscilatory motion in k -space, accompanied by a peri-odic motion in the real space is termed Bloch oscillations. The main issuewas whether or not this picture is washed out by the interband coupling( P H ε (1 − P ) + h.c. ). Wannier [3], [4] argued that one can redefine thebands of H so that the one-band Hamiltonian P ε H ε P ε where P ε is the orthogonal projection on the subspace of states correspondingto a deformed band, has again discrete spectrum and the non-diagonal partvanishes, P ε H ε (1 − P ε )+ h.c. = 0, i.e. the deformed bands are ”closed” underthe dynamics given by H ǫ . Unfortunately, as discussed in the Introduction,the existence of closed bands is ruled out by the fact that the spectrum of H ε is absolutely continuous.The main result in [1] is a recurrent rigorous construction of deformedbands σ n so that the interband coupling although nonzero are small, i.e. if3 εn is the orthogonal projection on the subspace of states corresponding tothe deformed band, then P εn H ε (1 − P εn )is of the order ε n +1 , n = 1 , , ... . This implies that γ n ( ε, t ) = k (1 − P εn ) e − iH ε t P εn k ≤ b n ε n +1 t (2.3)Taking into account that 1 − γ n ( ε, t ) is a lower bound for the probability offinding at time t the electron in a state corresponding to σ n if at t = 0 theelectron is with probability one in a state corresponding to σ n , it follows thatfor states corresponding to σ n and time scales of order t ≃ ε − n , the dynamicsgenerated by the full Hamiltonian H ε is well approximated by the dynamicsgenerated by the one-band Hamiltonian P εn H ε P εn .Coming back to our time dependent electric field problem, the Hamilto-nian of the system is H ε,ω ( t ) = H + εX F ( ωt ) (2.4)with F ( u ) and all its derivatives F ( n ) ( u ) bounded. The case F ( u ) = 1 is theone discussed above.Heuristically, it is expected by an adiabatic argument that for small ω the transitions caused by the time dependence of the electric field are stillsmall and one hope the same type of result. More precisely, if U ε,ω ( t ) is thesolution of the Schr¨odinger equation i dU ε,ω ( t ) dt = H ε,ω ( t ) U ε,ω ( t ) (2.5)we are looking for an operator P ε,ωn ( t ), n = 0 , , , ... , P ε,ω ( t ) = P , so thatthe interband transitions be bounded by γ n ( ε, ω, t ) = k (1 − P ε,ωn ( t )) U ε,ω ( t ) P ε,ωn ( t ) k≤ tε n X α =0 C α ε n − α ω α (2.6)A recurrent construction of P ε,ωn ( t ) such that (2.6) holds true is the mainresult of this paper.We end up this section with a few remarks.i. As expected, in the limit ω → P ε,ωn ( t ) is constructed out of H ε,ω ( t ) and its derivatives up to order n .iii. As in the time independent electric field case [1] the smallness ofinterband transitions implies the validity of one -band approximation on longtime scales (of order min α =0 , ,...n ( ε n − α ω α ) − , n = 1 , , ... ). However, since4oth H ε,ω and P ε,ωn depend on time, the analysis of the one-band dynamicsis more complicated than in the time independent electric field case [19] andis deferred to a future publication.iv. As already said in the Introduction, in [14], [15] we developed for theabove Hamiltonian (2.4) a similar theory as in the time independent caseup to the second order. More exactly, we redefined the deformed bands of H and for these deformed bands, in the second order theory the interbandtransitions are bounded by γ ( ε, ω, t ) ≤ ( C ε + C ε · ω ) | t | (2.7)The recurrent procedure was not developed further to an arbitrary order n ,the higher order construction implying very laborious calculations. P ε,ωn ( t ) In the following we shall use a procedure based on the adiabatic expansiontheorem developed in [16].Unfortunately, the Hamiltonian of the problem (2.4) is not of an adiabatictype. Moreover, in this problem we are dealing with two small parameters ε and ω .If we rescale s = εt ; ω = εa ; a − parameter the Schr¨odinger equation becomes: iε dU ε ( s, a ) ds = H ε ( s, a ) U ε ( s, a ) (3.1)Defining U ( s, a ) ≡ e − iX G ( s,a ) (3.2)where G ( s, a ) = Z s F ( au ) du (3.3)and W ε ( s, a ) ≡ U ∗ ( s, a ) U ε ( s, a ) (3.4)the Schr¨odinger equation becomes of the adiabatic form [16], but with anaditional parameter a : iε dW ε ( s, a ) ds = e H ( s, a ) W ε ( s, a ) (3.5)5here e H ( s, a ) = U ∗ ( s, a ) H U ( s, a ) (3.6)has the same spectrum as H .Now, in terms of W ε ( s, a ) the interband transitions (2.6) become [14]: γ n ( ε, ω, t ) = γ n ( ε, s, a ) ≡k (1 − e P εn ( s, a ) W ε ( s, a ) e P εn (0 , a ) k (3.7)where e P εn ( s, a ) = U ∗ ( s, a ) P ε,ωn ( t ) U ( s, a ) (3.8)have to be constructed. Once e P εn ( s, a ) constructed, P ε,ωn ( t ) are given by (3.8).At fixed a , the construction of e P εn ( s, a ) follows closely the method in [16] butemphasizing the a dependence.We define the sequence e E j ( s, a ) by the recurrence formula (see Lemma 1in [16]): e E ( s, a ) = e P ( s, a ) = i π I Γ e H ( s, a ) − z dz = i π I Γ e R ( s, a ; z ) dz (3.9) e E j ( s, a ) = − π I Γ e R ( s, a ; z )[(1 − e P ( s, a )) e E (1) j − ( s, a ) e P ( s, a ) − h.c. ] e R ( s, a ; z ) dz ++ e S j ( s, a ) − e P ( s, a ) e S j ( s, a ) e P ( s, a ) (3.10)where e S j ( s, a ) = j − X m =1 e E m ( s, a ) e E j − m ( s, a ) (3.11) e E ( n ) j ( s, a ) = d n e E j ( s, a ) ds n and Γ is a contour enclosing the isolated band σ . e E j ( s, a ) satisfy: e E j ( s, a ) = j X m =0 e E m ( s, a ) e E j − m ( s, a ) (3.12) i e E (1) j − ( s, a ) = [ e H ( s, a ) , e E j ( s, a )] (3.13)As a consequence of (3.12), (3.13), T εn ( s, a ), n = 0 , , , ... defined by: T εn ( s, a ) = n X j =0 e E j ( s, a ) ε j (3.14)6ave the properties: iεT ε (1) n − [ e H ( s, a ) , T εn ( s, a )] = i e E (1) n ε n +1 (3.15) k ( T εn ( s, a )) − T εn ( s, a ) k∼ O ( ε n +1 )Finally, following [16], [20] we construct projection operators e P εn ( s, a )corresponding to almost invariant subspaces describing the deformed bands: e P εn ( s, a ) = i π Z | z − | = ( T εn ( s, a ) − z ) − dz == T εn ( s, a ) + (cid:18) T εn ( s, a ) − (cid:19) n(cid:2) (cid:0) ( T εn ( s, a )) − T εn ( s, a ) (cid:1)(cid:3) − − o (3.16)The crucial property of e P εn ( s, a ) is: iε e P ε (1) n ( s, a ) − h e H ( s, a ) , e P εn ( s, a ) i == − ε n +1 π Z | z − | = ( T εn ( s, a ) − z ) − e E (1) n ( s, a ) ( T εn ( s, a ) − z ) − dz (3.17)Using the fact that (1 − e P εn ( s, a )) e P εn ( s, a ) = 0 and that k (1 − e P εn ( s, a )) k = k W ε ( s, a ) k = 1 the interband transitions (3.7) can be rewritten as: γ n ( ε, s, a ) = k (cid:16) − e P εn ( s, a ) (cid:17) W ε ( s, a ) e P εn (0 , a ) W ε ∗ ( s, a ) W ε ( s, a ) k = k (cid:16) − e P εn ( s, a ) (cid:17) h − e P εn ( s, a ) + W ε ( s, a ) e P εn (0 , a ) W ε ∗ ( s, a ) i W ε ( s, a ) k≤ (3.18) ≤ k e P εn ( s, a ) − W ε ( s, a ) e P εn (0 , a ) W ε ∗ ( s, a ) k It remains to estimate the last norm in (3.18). The main point is thatin order to obtain estimations of the form (2.6) one has to control the a dependence. We begin with a preparatory result.
Lemma 4.1. k e P εn ( s, a ) − W ε ( s, a ) e P εn (0 , a ) W ε ∗ ( s, a ) k≤ ε Z s k iε d e P εn ( u, a ) du − h e H ( u, a ) , e P εn ( u, a ) i k du (4.1)7 roof. The proof is standard [21],[16] but we give it for completeness.Rewrite the l.h.s. of (4.1) as: e P εn ( s, a ) − W ε ( s, a ) e P εn (0 , a ) W ε⋆ ( s, a ) == W ε ( s, a ) h W ε⋆ ( s, a ) e P εn ( s, a ) W ε ( s, a ) − e P εn (0 , a ) i W ε⋆ ( s, a )Using (3.5), the equation satisfied by the function f ( s, a ) = W ε⋆ ( s, a ) e P εn ( s, a ) W ε ( s, a ) − e P εn (0 , a )is iε df ( s, a ) ds = W ε ∗ ( s, a ) ( iε d e P εn ( s, a ) ds − h e H ( s, a ) , e P εn ( s, a ) i) W ε ( s, a )The solution of this equation is f ( s, a ) − f (0 , a ) = 1 iε Z s W ε ∗ ( u, a ) ( iε d e P εn ( u, a ) ds − h e H ( u, a ) , e P εn ( u, a ) i) W ε ( u, a ) du Since W ε ( s, a ) is unitary and f (0 , a ) = 0, Lemma 4.1 results immediately.As a result (3.18) becomes: γ n ( ε, s, a ) ≤ ε Z s k iε d e P εn ( u, a ) du − h e H ( u, a ) , e P εn ( u, a ) i k du (4.2)Now from (4.2), the property (3.17) of the projection operators e P εn ( s, a )and the fact that ([16], [20]) sup | z − | = k (cid:0) T ε n ( s, a ) − z (cid:1) − k is bounded uni-formly in s it results: γ n ( ε, s, a ) ≤ const.ε n sup ≤ u ≤ s k e E (1) n ( u, a ) k · s (4.3)and what is left is to obtain estimations of k e E (1) n ( u, a ) k .In what follows e R ( s, a ; z ) = ( e H ( s, a ) − z ) − , R ( z ) = ( H − z ) − and Γa contour enclosing σ . We shall prove first: Lemma 4.2. sup s ∈ R ,z ∈ Γ k e R ( n )0 ( s, a ; z ) k≤ n − X l =0 C l a l (4.4)8 roof. For n = 1 , ,
3, by a direct calculation using (3.2), (3.3) and (3.6)one obtains: e R (1)0 ( s, a ; z ) = iF ( as ) U ∗ ( s, a ) [ X , R ( z )] U ( s, a ) e R (2)0 ( s, a ; z ) = iaF (1) ( as ) U ∗ ( s, a ) [ X , R ( z )] U ( s, a )++ F ( as ) U ∗ ( s, a ) [[ X , R ( z )] , X ] U ( s, a ) e R (3)0 ( s, a ; z ) = ia F (2) ( as ) U ∗ ( s, a ) [ X , R ( z )] U ( s, a )++ aF (1) ( as ) F ( as ) U ∗ ( s, a ) [[ X , R ( z )] , X ] U ( s, a ) −− iF ( as ) U ∗ [[[ X , R ( z )] , X ] , X ]In general, one can see recurrently that e R ( n )0 ( s, a ) is a polynomial of degree n − a whose coefficients are products of F k ( as ), U ∗ ( s, a ), U ( s, a ) andmultiple commutators [[ ... [ R ( z ) , X ] , ..., X ]]. Since all these factors (for themultiple commutators see e.g. [22], [1]) are uniformly bounded in a , s and z the proof of lemma is finished.Finally the next lemma gives the necessary estimate of k e E (1) n ( u, a ) k : Lemma 4.3. k e E j ( s, a ) k≤ j − X l =0 e l a l (4.5) k e E (1) j ( s, a ) k≤ j X l =0 f l a l (4.6) Proof.
We shall prove by induction that e E j ( s, a ) is a finite sum of terms,each term is a multiple integral on Γ, the integrand being m Y k =1 e R ( α k )0 ( s, a ; z ) (4.7)where α k ≥ X k α k = 19n addition, e E (1) j ( s, a ) have the same form with X k α k = j + 1For j = 0 this is trivial since (see (3.9)): e E ( s, a ) = e P ( s, a ) = i π I Γ e R ( s, a ; z ) dz and e E (1)0 ( s, a ) = i π I Γ d e R ( s, a ; z ) ds dz Suppose that e E j ( s, a ) satisfies the induction hypothesis and we want toprove the same is true for j + 1.From (3.10) e E j +1 ( s, a ) contains two types of terms:- The first type is a multiple integral of terms containing e E (1) j ( s, a ) andresolvents of e H ( s, a ). According to the induction hypothesis the terms areof the form (4.7)where X k α k = j + 1- the second type of terms contains e S j +1 ( s, a ): e S j +1 ( s, a ) = j X m =1 e E m ( s, a ) e E j +1 − m ( s, a )and again from the induction hypothesis they are of the above form (4.7)with X k α k = m + j + 1 − m = j + 1It results that e E j +1 is a finite sum of terms, each term being a multipleintegral on Γ, with the integrant of the form m Y k =1 e R ( α k )0 ( s, a ; z )with α k ≥ X k α k = j + 110y the Leibnitz rule, the derivative e E (1) j +1 ( s, a ) is of the same form, butwith X k α k = j + 2This and Lemma 4.2 give (4.5) and 4.6 which finishes the proof.Plugging (4.6) into (4.3) one obtains that γ n ( ε, s, a ) ≤ ε n s X k C k a k and going back to the variables t and taking into account that a = ωε itresults γ n ( ε, ω, t ) ≤ εt n X k =0 C k ε n − k ω k which is the desired result. Acknowledgements.
I would like to thank G. Nenciu for suggesting me the adiabatic expan-sion formalism and for helpful discussions. This research was supported byCNCSIS under Grant 905-6/2007.
References [1] Nenciu A, Nenciu G 1981
J. Phus A: Math. Gen. Rev. Mod. Phys. Phys. Rev.
Rev. Mod. Phys. Solid State Phys.
J. Math. Phys. Commun. Math.Phys. J. Phys.
C15
J. Math.Phys. Helv. Phys. Acta J. Phus A: Math. Gen. Phys. Rev. Lett. Romanian Rep. Phys. UPB Sci. Bull. Series C [16] Nenciu G 1993 Commun. Math. Phys.
Phys. Rev.
Quantum Theory of the Solid State (Academic Press,New York)[19] Nenciu A, Nenciu G 1982
J. Phus A: Math. Gen. J. Math. Phys. Perturbation Theory for Linear Operators (Springer,Berlin)[22] Avron J 1979
J. Phys. A: Math. Gen.12